• The theta correspondence 1 is a correspondence between automorphic forms on two members of a dual reductive pair. Because of the relation between automorphic forms and automor-phic representations, it is also a correspondence between automorphic representations. The Siegel-Weil formula says that certain natural linear combinations of theta lifts are Siegel-type holomorphic Eisenstein Series.

We present a level raising result for families of p-adic automorphic forms for a definite quaternion algebra D over Q. The main theorem is an analogue of a theorem for classical automorphic forms due to Diamond and Taylor. One of the ingredients in the proof of Diamond and Taylor’s theorem (which also played a role in earlier work of Taylor) is the definition of a suitable pairing on the space ...

Two of the primary methods of constructing automorphic forms are the Langlands program and Howe's theory of dual pairs. The Langlands program concerns a reductive linear group G deened over a number eld. Associated to G is its dual group L G. The con-jectural principle of functoriality says that a homomorphism L H ! L G should provide a \transfer" of automorphic representations from H to those ...

This is an attempt to formulate a geometric Langlands conjecture for G = GSpin2n+1 and GSp2n, n ≥ 1. Namely, let Ǧ be the Langlands dual group (over Q̄l), let X be a smooth projective connected curve. Given a Ǧ-local system E on X , assuming some irreducibility type conditions on VE for some representations V of Ǧ, we propose a conjectural construction of a distinguished E-Hecke automorphic shea...

We consider the automorphic forms which govern the gravitational threshold correction F1 in models of heterotic/IIA duality with N = 2 supersymmetry in four dimensions. In particular we derive the full nonperturbative formula for F1 for the dual pair originally considered by Ferrara, Harvey, Strominger and Vafa (FHSV). The answer involves an interesting automorphic product constructed by Borche...

We prove the compatibility of the local and global Langlands correspondences at places dividing l for the l-adic Galois representations associated to regular algebraic conjugate self-dual cuspidal automorphic representations of GLn over an imaginary CM field, under the assumption that the automorphic representations have Iwahori-fixed vectors at places dividing l and have Shin-regular weight. 2...

Let A be the ring of adeles a number field F. Given self-dual irreducible, automorphic, cuspidal representation ? GLn(A), with trivial central character, we construct its full inverse image under weak Langlands functorial lift from appropriate split classical group G. We do this by new automorphic descent method, namely double descent. This method is derived recent generalized doubling integral...

An exercise in Garrett Birkhoff’s renowned book on lattice theory asks for a lattice with 18 elements and of length 5 that has a dual automorphism, but no involutory dual automorphism. This note constructs a smaller lattice, 15 elements and length 4, with the same property.

We generalize the construction of dual BGG complexes in Faltings–Chai and Mokrane–Tilouine–Polo (from the case of Siegel modular varieties) to all smooth integral models of PEL-type Shimura varieties.

We show the uniqueness and disjointness of Klyachko models for GLn over a non-archimedean local field. This completes, in particular, the study of Klyachko models on the unitary dual. Our local results imply a global rigidity property for the discrete automorphic spectrum.